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Robust Continuous-Time Smoothers Without Two-Sided Stochastic Integrals Vikram Krishnamurthy, Senior Member, IEEE, and Robert Elliott

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Robust Continuous-Time Smoothers Without Two-Sided Stochastic Integrals Vikram Krishnamurthy, Senior Member, IEEE, and Robert Elliott 1824 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 11, NOVEMBER 2002 Robust Continuous-Time Smoothers Without Two-Sided Stochastic Integrals Vikram Krishnamurthy, Senior Member, IEEE, and Robert Elliott Abstract—We consider the problem of fixed-interval smoothing locally Lipschitz continuous in the observations, i.e., the equa- of a continuous-time partially observed nonlinear stochastic tions depend continuously on the observation path. Indeed, the dynamical system. Existing results for such smoothers require equations turn out to be nonstochastic parabolic partial differ- the use of two-sided stochastic calculus. The main contribution of this paper is to present a robust formulation of the smoothing ential equations whose coefficients depend on the observations. equations. Under this robust formulation, the smoothing equa- Apart from not requiring the intricacies of two-sided stochastic tions are nonstochastic parabolic partial differential equations calculus, these robust equations are useful from a practical point (with random coefficients) and, hence, the technical machinery of view; their numerical solution via time discretization can be associated with two sided stochastic calculus is not required. performed without worrying about the Ito terms. Furthermore, the robust smoothed state estimates are locally Lipschitz in the observations, which is useful for numerical The idea of robust filtering, i.e., re-expressing the stochastic simulation. As examples, finite dimensional robust versions of the differential equation as nonstochastic differential equation with Benes and hidden Markov model smoothers and smoothers for random coefficients has been used extensively in the context piecewise linear dynamics are derived; these finite-dimensional of nonlinear filtering; see, for example, [6], [16], [8], [18], or smoothers do not involve stochastic integrals. [2, Ch. 4]. More recently, in [14], versions of these robust fil- Index Terms—Continuous time, hidden Markov models ters, probabilistic interpretations and implicit and explicit dis- (HMMs), maximum likelihood estimation, nonlinear smoothing, cretization schemes were developed for continuous-time hidden piecewise linear models, stochastic differential equations. Markov models (HMMs). The contributions of this paper are as follows. I. INTRODUCTION 1) It is shown in Section III that the smoothed state estimate ILTERING is another word for conditional mean estima- can be computed via robust forward and backward filters. F tion of the state at time of a given dynamical stochastic Each of these filters involve nonstochastic parabolic par- system, based on the available incomplete information (obser- tial differential equations. vations) until the same time . Fixed-interval smoothing refers 2) Robust fixed interval smoothed estimates of functionals to the problem when given a trajectory of observations up to of the state of the system are derived in Section III. some fixed time , one wishes to compute the conditional Again, the equations involve nonstochastic integrals. mean estimate of the underlying state at times in the interval These robust smoothers can be used in maximum . likelihood parameter estimation via the expectation For continuous-time dynamical stochastic systems, the fil- maximization (EM) algorithm. The EM algorithm (see tered state density can be expressed as a stochastic partial dif- Section II-B) is a widely used numerical method for ferential equation called the Duncan–Mortenson–Zakai (DMZ) computing the maximum likelihood parameter estimate equation [2]. Derivation of the fixed-interval smoothed state for partially observed stochastic dynamical systems; density is mathematically more formidable as it requires the use see, for example, [23], [4], and [14]. Unlike this paper, of two sided stochastic calculus [19]. in [14] and [9], two-sided stochastic calculus involving In this paper we derive robust filters and smoothers for the Skorohod and generalized Stratonovich integrals are state of a continuous-time stochastic dynamical system by using used to derive smoothers for computing estimates of the a gauge transformation, see for example [6], [8]. By robust we functionals required in the EM algorithm for HMMs and mean that the resulting filtering and smoothing equations are linear Gaussian state space models, respectively. 3) As examples of the robust smoothers for the state and functionals of the state, we present state and maximum Manuscript received July 18, 2001; revised March 9, 2002, April 18, 2002, and July 21, 2002. Recommended by Associate Editor T. Parisini. This work likelihood parameter estimation for three classes of sto- was supported by the Australian Research Council and the NSERC, Canada. chastic dynamical systems: 1) Benes type nonlinear dy- V. Krishnamurthy is with the Department of Electrical and Computer En- namical systems with non Gaussian initial conditions (see gineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada (e-mail: [email protected]), and also with the University of Melbourne, Mel- Section IV), 2) HMM (see Section V), and 3) systems boune, Australia. Portions of this work were conducted while the author was on with piecewise linear dynamics (see Section VI). sabbatical at the Department of Signals Sensors and Systems, Royal Insitute of Technology, 100 44 Stockholm, Sweden. Instead of using fixed-interval smoothing for cases 1) and 2), R. Elliott is with the Haskayne School of Business, University of Calgary, finite-dimensional filters have been derived in [12], [13], and Alberta, AB T2N 1N4, Canada (e-mail: [email protected]). Portions of this [14] to compute estimates of the functionals required in the EM work were completed while the author was visiting the Department of Applied Mathematics, University of Adelaide, Australia. algorithm. However, the computational complexity of these fil- Digital Object Identifier 10.1109/TAC.2002.804481 ters are for some of the functionals (e.g., for the number 0018-9286/02$17.00 © 2002 IEEE KRISHNAMURTHY AND ELLIOTT: ROBUST CONTINUOUS-TIME SMOOTHERS 1825 of jumps in an HMM) at each time instant where denotes the tinuous functions on [0, ]). Also state dimension. In comparison, computing estimates of these endowed with the sup-norm, i.e., functionals via fixed-interval smoothers involves a complexity . of but requires storage memory of where is the We also assume throughout that for all , A5) length of the observation data sequence. Approximate filtering holds. for piecewise linear systems via a bank on Kalman filters is pre- A5) , and are continuously differentiable with re- sented in [20] and [21]. We extend these results to derive robust spect to the parameter . The derivatives and smoothers for the state and functionals of the state required in are measurable and bounded functions. the EM algorithm, see Section VI for details. To introduce the gauge transformation we shall as- sume A6). II. MODEL AND PROBLEM FORMULATION A6) has continuous and bounded first and second derivatives with respect to and bounded first deriva- A. Signal Model and Objectives tive with respect to . The differentiability w.r.t. is not Consider the following continuous-time partially observed required in the finite-state Markov case considered in nonlinear stochastic dynamical system defined on the measur- Section V. able space ( , ). Let { }, where denotes a com- In Section VI, the assumption of continuous first and second pact subset of , denote a family of parametrized probability derivatives is relaxed. In particular Section VI assumes that measures. Under , the state { } taking values in , and is piecewise linear and continuous in . Tanaka’s for- the observation process { } taking values in , are de- mula, which is roughly speaking an extension of Ito’s formula scribed by to the nondifferentiable case, will be used. Objectives : In this paper, we will derive robust filtering and (1) smoothing equations. By robust, we mean that the solution to (2) the resulting equations are locally Lipschitz continuous in the observation . As mentioned in Section I, this is a useful prop- Let denote a fixed real number. For , define the erty from an implementation point of view. The aim of this paper right-continuous filtrations { }, { }, and { } with is threefold. i) Derive robust fixed-interval smoothers for that do not involve stochastic integrals. ii) Derive robust fixed interval smoothers for functionals of (3) the form In (1) and (2), { } and { } are independent standard Brownian motions. Further, { } and { } are independent of . (In Section V, we will consider the HMM case where is (4) a measurable finite state zero mean martingale process. where , , We make the following standard assumptions [2, pp. 114] for are Borel measurable and bounded functions. all . is assumed once differentiable in . Our aim is to com- pute the fixed-interval smoothed estimate , A1) and using robust forward and backward filters. are bounded Borel measurable functions These smoothed estimates are required in computing the A2) is continuous and bounded maximum likelihood parameter estimate via the EM al- such that is a uniformly positive definite gorithm; see Section II-B. The same problem is consid- matrix, i.e., for some real . This ered in [4] where two-sided stochastic calculus was used ensures that the backward operator (defined in (15)) to compute . is uniformly elliptic. This condition can be somewhat To motivate the robust smoothers presented below, relaxed with replaced by its pseudoinverse , consider
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